A Study of Enfortumab Vedotin in People With Adenoid Cystic Carcinoma

Inclusion Criteria: * Patients must have pathologically or cytologically confirmed adenoid cystic carcinoma. Cancers arising from non-salivary gland primary sites are allowed. * Patients must have recurrent and/or metastatic disease not amenable to other curative intent therapy. * At least 4 weeks must have elapsed since the end of prior systemic treatment and/or 2 weeks since completion of radiotherapy with resolution of all treatment related toxicity to NCI CTCAE Version 5.0 grade \<1 (or tolerable grade 2) or back to baseline (except for alopecia, lymphopenia, or hypothyroidism) prior to starting study drug treatment. * Patients must have RECIST V1.1 measurable disease, defined as at least one nonnodal lesion measuring ≥ 20 mm with conventional techniques or as ≥10mm with CT scan, MRI, or calipers by clinical exam in the longest dimension AND/OR a nodal lesion measuring ≥ 15 mm in the shortest dimension. Tumors in previously irradiated fields may be considered measurable if there is evidence of tumor progression after radiation treatment. * Patients must have documentation of a new or progressive lesion on radiologic imaging study performed within 6 months prior to study enrollment (progression of disease over any interval is allowed) and/or new/worsening disease related symptoms within 6 months prior to study enrollment. Note: This assessment will be performed by the treating investigator and evidence of progression by RECIST criteria is not required. * Age ≥ 18 years of age on the day of signing informed consent. * ECOG performance status 0 or 1 (or Karnofsky ≥ 70%). * Patients must have tissue from the primary tumor or metastases available for correlative studies. Either a paraffin block or at least 20 unstained slides are acceptable (paraffin block or at 30 unstained slides would be ideal). Patients without available tissue for submission may still be eligible if approved by the Principal Investigator. Additional tissue collection is not a requirement for this study. * Screening laboratory values must meet the following criteria: * Neutrophils ≥ 1500/μL * Platelets ≥ 100x10\^3/μL * Hemoglobin \> 9.0 g/dL (without packed red blood cell (pRBC) transfusion within the last 2 weeks) * AST and ALT ≤ 2.5 x ULN (if liver metastases are present, AST and ALT ≤ 5x ULN) * Total Bilirubin ≤ 1.5 x ULN OR direct bilirubin ≤ ULN for participants with total bilirubin levels \>1.5 x ULN (except participants with Gilbert Syndrome, who can have a total bilirubin \< 3.0 mg/dL) * Serum creatinine ≤ 1.5 x ULN OR creatinine clearance (CrCl) ≥ 40 mL/min per the Cockcroft-Gault formula if creatinine is \>1.5 x ULN * Female CrCl = (140 - age in years) x weight in kg x 0.85 72 x serum creatinine in mg/dL * Male CrCl = (140 - age in years) x weight in kg x 1.00 72 x serum creatinine in mg/dL * Participants must be willing to sign the written informed consent form. A signed informed consent form must be appropriately obtained prior to the conduct of any trial specific procedure. * Male participants must agree to use adequate contraception and refrain from donating sperm from start of therapy through 4 months after last dose of trial treatment. * Female participants must agree not to donate ova starting at screening and throughout the study period, and for at least 3 weeks after the final dose of study drug. * A female participant is eligible to participate if 1) she is not pregnant (for women of child-bearing potential, a pregnancy test must be negative within 72 hours prior to initiation; if a urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required) OR 2) she is not a woman of child bearing potential as defined by one of the following criteria: * Pre-menopausal with one of the following: documented hysterectomy, documented bilateral salpingectomy, documented bilateral oophorectomy * Postmenopausal females defined as no menses for 12 months without an alternative medical cause. However, in the absence of 12 months of amenorrhea, confirmation with two FSH measurements in the postmenopausal range is required. * A woman of childbearing potential must use highly effective contraception from the start of therapy through 2 months after the last dose of study medication. Exclusion Criteria: * Untreated brain metastasis (subjects with treated brain metastases will be eligible, provided that they are radiographically stable, i.e. without evidence of progression for at least 4 weeks by repeat imaging \[note that the repeat imaging should be performed during study screening\], clinically stable and without requirement of steroid treatment for at least 14 days prior to the first dose of study treatment). * Prior malignancy if diagnosed and treated within 2 years of trial drug initiation. Patients may be included if they have completed therapy for a prior malignancy \>2 years prior to drug initiation and are currently no evidence of disease (NED). Exception: Participants with non-melanoma skin cancer or carcinoma in situ (breast DCIS, or cervical CIS) that have undergone potentially curative therapy at any time are not excluded from trial participation. * Prior systemic anti-cancer therapy within 4 weeks of start of study treatment. * ≥ Grade 2 peripheral neuropathy per CTCAE v5.0 criteria. * Dry eyes, keratitis, keratopathy, and active conjunctivitis. * New York Heart Association congestive heart failure of grade II or above, unstable angina, myocardial infarction within the past 6 months, or serious cardiac arrhythmia associated with significant cardiovascular impairment within the past 6 months (baseline echocardiogram is not required unless clinically indicated). * Active infection, defined as any infection requiring systemic treatment * Subject is known to be positive for Human Immunodeficiency Virus (HIV) or active Hepatitis C Virus (HCV) or active hepatitis B (HBV) infection (positive viral load). Testing for HIV, HCV, or HBV prior to initiation of the study drug is not required. If patient's have a known history of treated HCV, then a viral load is required to confirm clearance of infection. * Has a history of (non-infectious) pneumonitis/interstitial lung disease that required steroids or has current pneumonitis/interstitial lung disease. * Renal failure requiring active hemo- or peritoneal dialysis. * Breast feeding is not allowed from the start of treatment through 3 weeks after the last dose of study drug. * Has a history or current evidence of any medical or other condition, therapy or laboratory abnormality which, in the opinion of the investigator, might confound the results of the study, or preclude participation in a clinical study.